李群在$\mathbb{R}^n$和框架上的连续作用
Continuous action of Lie groups on $\mathbb{R}^n$ and Frames
摘要：
在过去的十年里，小波和框架已经成为数学、物理和应用科学中广泛使用的工具。本文讨论了利用闭子群$H\subset \mathrm{GL}(n,\mathbb{R})$的作用构造$L^2(\R^n)$的框架，使得$H$在作用$(h,ω)\mapsto (h^{-1})^T(ω)$下在$\R^n$中有一个开轨道$\cO$.如果$H$有形式$ANR$，其中$A$是单连通的和阿贝尔的，$N$包含一个余紧的离散子群和$R$是紧的，包含$ω\in\cO$的稳定群，那么我们构造了一个框架的空间$L^2_{\cO}(\R^n)$的$L^2$ -函数，其傅里叶变换是支持在$\cO$。我们将其应用到的情况下，$H^T=H$和稳定是一个对称的子群，讨论的连续小波变换的情况下，由Fabec和Olafsson的一份文件。
英文摘要：
Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\R^n)$ using the action of closed subgroups $H\subset \mathrm{GL}(n,\mathbb{R})$ such that $H$ has an open orbit $\cO$ in $\R^n$ under the action $(h,ω)\mapsto (h^{-1})^T(ω)$. If $H$ has the form $ANR$, where $A$ is simply connected and abelian, $N$ contains a co-compact discrete subgroup and $R$ is compact containing the stabilizer group of $ω\in\cO$ then we construct a frame for the space $L^2_{\cO}(\R^n)$ of $L^2$-functions whose Fourier transform is supported in $\cO$. We apply this to the case where $H^T=H$ and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in a paper by Fabec and Olafsson.
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